#235378

order	512
nilpotency class	3
isoclinism	isoclinic class of #235369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,23,27)(19,28,32,43)(20,25,34,42)(21,26,22,35)(24,30,38,31)(29,44,46,48)(33,40,36,41)(37,45,39,47);(1,3,6,13)(2,7,10,15)(4,14,5,12)(8,16,9,11)(17,19,23,32)(18,24,27,38)(20,29,34,46)(21,31,22,30)(25,39,42,37)(26,43,35,28)(33,47,36,45)(40,44,41,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,40)(27,42)(28,44)(30,45)(31,47)(32,46)(35,41)(38,39)(43,48);(17,21,23,22)(18,26,27,35)(19,30,32,31)(20,33,34,36)(24,28,38,43)(25,40,42,41)(29,45,46,47)(37,44,39,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142574);